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Questions tagged [mgf]

The moment generating function (mgf) is a real function which allows to derive a random variable's moments and therefore can characterize its entire distribution. Use also for its logarithm, the cumulant generating function.

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Why can we treat MGF in this way

For the standard proof that if $Z \sim N(0,1)$ than $Z^{2} \sim \chi^{2}_{1}$ We write: $$M_{Z^{2}}(t)=\int_{\mathbb{R}}\exp(tz^{2})\frac{1}{\sqrt{2\pi}}\exp(\frac{-z^{2}}{2}) dz$$ That is, we use ...
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Moment Generating Function for Lognormal Random Variable

I'm working through the proof of a lognormal random variable and am having some difficulty in moving through it. I understand the following: Our CDF is $\Phi(\frac{logx - \mu}{\sigma})$, and thus our ...
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Sufficient conditions for multivariate MGF to be finite in neighborhood of zero

In the one-dimensional case, we have the following fact: (see Existence of the moment generating function and variance for proof) Proposition: The mgf $m(t)$ is finite in an open interval $(t_n,t_p)$ ...
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Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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Relations between moments and cumulants

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= ...
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Moment generating function between 2 variables

I understand that mean of Y is M'Y(0), whereas variance of Y is M''Y(0). I can derive expressions through differentiation to get M'Y(0) and M''Y(0). The 2, however, have the expression Mx(0) in them. ...
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How does MGF of Pareto distribution of first kind exist for non-positive values of t? [closed]

I have reached upto the stage shown in the attached picture. The r.v. X is always positive and its power $\beta+1$ is also always positive. Therefore, how can it be said that MGF exists for t <= 0? ...
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Establishing an upper bound for the tail probability $P(X-\lambda \geq z)$ for any $z>0$, where $X$ is Poisson r.v. w/ parameter $\lambda$

Poisson random variable $X$ with the parameter $\lambda$ has, respectively, the pmf and the moment generating function of the forms $$P(X = k) = \dfrac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\...
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In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
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Sum of negative binomial variables with transformed parameters

Let $X_i:i=1,2,...,n$ be independent ~ $NegBin(\mu,\alpha)$random variables such that $E(X_i) = \mu$, $Var(X_i) = \mu + \frac{\mu^2}{\alpha}$. (i) Find the mean and variance of $Y=∑(X_i)$. (ii) Find ...
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Tail bound for sum of i.i.d. random variables with common moment generating function

Suppose $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of independent and identically distributed random variables and $S_n:=X_1+...+X_n$. Assume that each $X_i$ has mean $0$ and that all $X_i$ have a ...
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Trying to Understand Power Series Expansion of Cumulant Generating Function

I'm having difficulty expanding the cumulant generating function, $K_X(t)$ as a power series. Specifically, I'm trying to go from the definition of the cumulant generating function, i.e. $$K_X(t) = \...
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Help on Moment Generating Functions

I have recently been given a set of practice problems for my probabilities course and I have no idea where to even start on this question. The distribution of X = the number of toppings ordered by ...
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sum of $N$ gamma distributions with $N$ being a poisson distribution

I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution. I $N$ number of events start in $t$ minutes, the what will ...
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Moment generating function of a Weibull distribution and root finding heavy and light tailed case

I consider the equation $M_x(v)=1+(1+\beta)\mu$ and I need to find the solution $v>0$ such that the equation is fulfilled. For this example I consider the moment generating function $M_X(v)$ of a ...
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MGF of inverse discrete random variable

The following question is straightforward to show with Markov's Inequality: Let $X_n$ be uniformly distributed on the set {1, 2, ..., n}, and let $Y_n = 1/X_n$. Prove that $Y_n \overset{p}{\...
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Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
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Finding the distribution and its parameters with moment generating function

I am learning basic statistics and I am trying to solve the (example) problems but I can't figure out how to solve the following problem. I understand how to use the MGF to find expected value and ...
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correlation out of moment generating function [duplicate]

Suppose I have two random variables $X$ and $Y$ and their common moment generating function $Z(x,y)$. How can I use this to determine the correlation of $X$ and $Y$? It's supposed to be something ...
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MGF of the multivariate hypergeometric distribution

Does the multivariate hypergeometric distribution, for sampling without replacement from multiple objects, have a known form for the moment generating function?
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Sum of normal independent random variables with coefficients

I'm trying to wrap my head around linear transformations to random variables (with coefficients > 1). Consider the two random and independent variables $X$ and $Y$ where: $$X \sim \mathcal{N}(0,1)\...
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In the Poisson Gamma model, how do you show the random variables in the sample are unconditionally dependent?

The poisson-gamma model has $n_i |\lambda$ ~ Poiss($\lambda$), with $\lambda$ ~ $gamma(a,b)$. (Prior) I know that ultimately you can show that unconditionally $n_i$ ~ NegBin($a, (b+1)^{-1}$). And ...
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Simulation of a random variable given the moment generating function after exponential tilt

The random variable $S$ follows a distribution with moment generating function $$M_S(v)=\frac{\beta\mu v}{1+(1+\beta)\mu v-M_X(v)}$$ I have been looking in some books about this m.g.f and I found ...
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Showing that two Brownian Motions are equal in distribution

I must show that $\{B(ct), t\geq 0\}$ is equal in distribution to $\{c^{1/2}B(t), t\geq 0\}$ where $B(t)$ is a Brownian Motion and $c$ is some constant. So, I'll be honest. I'm at a loss. I've tried ...
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property of the normalising constant in an exponential family

A statistical model for the data set $\bf{y}$ is an exponential family with canonical parameter vector $\theta = (\theta_1,.. \theta_k)$ and canonical statistic $\bf{t(y)} $=$(t_1(\boldsymbol{y}),.....
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By conditioning on $N$, show that the moment generating function of $Y$ is given by $m_Y(t)=m_N(\ln(m_X(t)))$

I am having a difficult time using moment generating function properties to prove this: (any direction or key properties will be very helpful) Let $X_1$, $X_2$, . . . be independent and identically ...
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The distribution of the product of a multivariate normal and a lognormal distribution

If $$X=\left(\begin{array}{c} X_{1}\\ X_{2} \end{array}\right)\sim N\left[\left(\begin{array}{c} \mu_{X_{1}}\\ \mu_{X_{2}} \end{array}\right),\left(\begin{array}{cc} \sigma_{X_{1}}\\ \sigma_{X_{1}X_{2}...
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Poisson and Negative Binomial distributions- Mean and Variance total claim size

I am trying to do the following: Let $S$ be the total claim size when the number of claims follow a Negative Binomial Distribution. How can I derive a formula for the E(S)- expectancy and V(S)-...
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Moment generating functions of continous random variables at 0

I'm studying mgfs of various distributions and I have a doubt about a property of mgfs: My book says that a common feature of discrete function's mgf is that it always exists, and takes value 1 at t=0 ...
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Find mean and variance of random variable using moment generating function

Suppose $X \sim U(a,b)$. Find the mean and variance of $X$. I tried to solve the exercise by computing the moment generating function and then substitute 0 to the ts but it doesn't work. How should ...
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Does inverse of a Gamma moment generating function have a known distribution?

I have come across a moment generating function for a random variable $Y$ of the following form $$M_Y(t) = \mathbb E\left[e^{tY}\right] = \left[1 - \frac t \beta\right]^k.$$ So it is basically $[M_X(t)...
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Derive Expectation/Variance of Histogram

Derive the expectation and variance of the following: $\frac{1}{n(b-a)}$$\sum_{i=1}^n 1$$_{Y_i∈(a,b]}$, b>a. My thoughts with this are to put the summation in an integral and then multiply by n. ...
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Calculating the m.g.f. of a Linear Function

I'm having trouble finding out how to determine the mgf of a linear transformation. The problem I have gives me an mgf in the form of $\exp(t^2+ct)$, where $c$ is a constant that I'm not mentioning. ...
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Numerical method to compress empirical probability distribution

I am trying to grapple with the following problem. I have an application that develops empirical distributions. In essence, I end up with a histogram of equally spaced $x$ values, with both a $max$...
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Expectation of square root of sum of independent squared uniform random variables

Let $X_1,\dots,X_n \sim U(0,1)$ be independent and identicallly distributed standard uniform random variables. $$\text{Let }\quad Y_n=\sum_i^nX_i^2 \quad \quad \text{I seek: } \quad \mathbb{E}\big[\...
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Saddlepoint approximation with weibull distribution

I have some trouble with this computation, I have the moment generating function of a random variable $S$ by: $$M_S(t)=\frac{\beta\mu t}{1+(1+\beta)\mu t-M_X(t)}$$ According to the text that I am ...
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Skewness of Tweedie distribution

Tweedie distributions are a family of distributions from the exponential dispersion family that have power-law mean-variance relationship: \begin{align} \mathbb E[X] &= \mu \\ \operatorname{Var}[...
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How to find the support of MGF

I am having a difficult time determining the support or range of a MGF for a given pmf. The specific questions states to find the MGF of f(x)=6/((x^2)(pi^2)) for x=1,2,3... The result (which is the ...
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Motivation behind the definition of a heavy-tailed distributions

The current Wikipedia definition is The distribution of a random variable $X$ with distribution function $F$ is said to have a heavy (right) tail if the moment generating function of $F,$ $MF(t),$ ...
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To find the probability when a function of normal variate is given

If $Y= \log10{(X)}$ .Y follows $N(\mu,\sigma^2)$ and has the mgf $My(t)=e^(5*t+2*t^2) Then P(X<1000)=? The far I got is that after comparing the given mgf with that of normal we get mu=5 and sigma^...
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How to solve moment generating function for attaining probability?

Mx(t) = 1/216*(5+e^t)^3 t€R Then P(X>1)=? The far I got it is that..It coincides with the mgf of binomial distribution and after comparing we get...... q=5. p=1. n=3. ...
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Cumulant generating function

I have a question, I consider the cumulant generating function given by the definition: Let $X\in R^d$ $$K_X(v)=\ln E[\exp(v^T X)].$$ If I have a random variable defined as: $$S=\sup_{t\geq 0}L_t$$ ...
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Probability distribution from momenta

I would like to get the probability distribution (either pdf or cdf) for a variable, by knowing the first n-momenta of the distribution. I ask: Is there a standard way to deal with this, and maybe a ...
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Real life uses of Moment generating functions

In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in ...
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For independent RVs $X_1,X_2,X_3$, does $X_1+X_2\stackrel{d}{=}X_1+X_3$ imply $X_2\stackrel{d}{=}X_3$?

Let $X_1,X_2$, and $X_3$ be independent random variables such that $X_1+X_2$ and $X_1+X_3$ have the same distribution. Does it follow that $X_2$ and $X_3$ have the same distribution? Can this be ...
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Non-existence of moment generating function

If moment generating function $E[e^{tX}]$ exists only for $t=0$, does it imply that every moment of $X$ does not exist(is not finite)?
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Central Limit Theorem and Moment Generating Functions

One can identify identical distributions by the equality of their Moment Generating Function (MGF), if it exists. Let's say $X_i$ ($i>0$) are independent random variables drawn from an identical ...
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What is the difference between moment generating function and probability generating function?

I am confused between the two terms " probability generating function" and "moment generating function." How do those terms differ?
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Does the trick of multiplying moment generating functions for the sum of variables work when the distributions are not identical?

If we have two random Variables that are independent and identically distributed, their sum's moment generating function is simply their product. But does the trick work if they are not identically ...